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The gauge is maxed out while it is cold. That makes me wonder if there is a problem either with the wiring or if the sending unit is bad. I am not sure if by it reading high all the time if there is an open circuit.
If you have a factory service manual (FSM) you can check the temp sensor by removing it and placing it in a pan of hot water. The service manual will give ohm readings for a given water temp. Either that or just go to O'-Reilies and get a temp sender.... I can look here in a few and tell you where that sensor is located.
"This has 55 ohms resistance at 260F (hot coolant) and it's resistance increases at lower temperature. Resistance is approximately 1400 ohms at 100F. This causes the current through the sender and and one small coil to increase as the coolant temperature increases. This moves the pointer."
On page 6A1B-5, item #169 is the "engine coolant temperature gage".
Item 169 is located right above the oil filter on the engine block. Looks like it's more to the rear of the filter and above the filter boss. You would probably mistake it for an oil pressure sensor. Might make sure the sensor wire is still connected before buying another sensor.
Guess they changed the location of the Sender - for every GM I've worked on, it's in the right head between 6 an 8 (V8). Oil Temp is at the filter. Both senders are basically a thermistor - resistance decreases with temperature. Anyway, disconnected (infinite resistance), your gage should peg low . With the signal wire grounded (no resistance), it should peg max. Yours has a short to ground somewhere or the Display is faulty.
On my '93 (assume is same on '96) both analog gauges always read hotter than the analog gauges and can the coolant temp can read as much as 30-40 degrees higher. I believe is there is a known issue with the accuracy or lack of with the analog gauges.
The ECT (Enginer Coolant Temp) sensor is on the left front of the engine just the underside of the waterpump housing.
The Engine Temp sensor (gage overheat) is on the rigth side between the back two cyliners.
Engine oil temp is on the left rear in the oil filter housing.
Because of where these sensors are located. I would say they are both accurate in their readings. Although cylinder heads absorb most of the combustion chamber heat, because of the reverse flow of coolant the cylinder heads receive coolant from the radiator before it flows to the block. Which is a good thing. Therefore if the digital sensor is located down by the oil filter...my first post would be inaccurate, plus the sensor would see higher temperatures.